Front. Earth Sci. 2014, 8(3): 439–456
DOI 10.1007/s11707-014-0416-0
RESEARCH ARTICLE
Artificial Neural Network (ANN) and Regression Tree (CART)
applications for the indirect estimation of unsaturated soil
shear strength parameters
D.P. KANUNGO (✉), Shaifaly SHARMA, Anindya PAIN
Geotechnical Engineering Group, CSIR – Central Building Research Institute (CBRI), Roorkee 247667, India
© Higher Education Press and Springer-Verlag Berlin Heidelberg 2014
Abstract The shear strength parameters of soil (cohesion
and angle of internal friction) are quite essential in solving
many civil engineering problems. In order to determine
these parameters, laboratory tests are used. The main
objective of this work is to evaluate the potential of
Artificial Neural Network (ANN) and Regression Tree
(CART) techniques for the indirect estimation of these
parameters. Four different models, considering different
combinations of 6 inputs, such as gravel %, sand %, silt %,
clay %, dry density, and plasticity index, were investigated
to evaluate the degree of their effects on the prediction of
shear parameters. A performance evaluation was carried
out using Correlation Coefficient and Root Mean Squared
Error measures. It was observed that for the prediction of
friction angle, the performance of both the techniques is
about the same. However, for the prediction of cohesion,
the ANN technique performs better than the CART
technique. It was further observed that the model
considering all of the 6 input soil parameters is the most
appropriate model for the prediction of shear parameters.
Also, connection weight and bias analyses of the best
neural network (i.e., 6/2/2) were attempted using Connection Weight, Garson, and proposed Weight-bias
approaches to characterize the influence of input variables
on shear strength parameters. It was observed that the
Connection Weight Approach provides the best overall
methodology for accurately quantifying variable importance, and should be favored over the other approaches
examined in this study.
Keywords cohesion, friction angle, Artificial Neural
Network, Regression Tree, Connection Weight, Weightbias Approach
Received May 15, 2013; accepted September 25, 2013
E-mail: debi.kanungo@gmail.com
1
Introduction
Shear strength parameters are the most important engineering properties of soil. Determinations of the shear
strength properties of soil are of foremost importance in
geotechnical investigation. Shear strength properties are
further required for the determination of bearing capacity
for foundation analysis. Understanding of the shear
strength of the soil is critical for the design of road
embankments, retaining walls, copper dams, etc. Shear
strength is the property of the soil that enables the soil to
keep its equilibrium when its surface is not level, or for that
matter, under any loading situation that produces shearing
stresses. Several procedures have been proposed in the
literature to determine the shear strength parameters of
unsaturated soil. These shear parameters can be determined
either in the field, or in the laboratory, or both. The tests
employed in the laboratory may include an unconfined
compression test, a triaxial test, a laboratory vane shear
test, a direct shear box test, and a direct simple shear test.
In-situ tests are normally conducted to confirm the validity
of the laboratory tests, and for design purposes. The in-situ
tests include a field vane shear test, a standard penetration
test, a cone penetration test, and piezocone and pressure
meter readings (Jain et al., 2010). Different authors have
worked considerably in the field of the prediction of shear
strength parameters of unsaturated soils, using mathematical relationships such as elliptical and hyperbolic
functions (Abra-mento et al., 1989; Escario and Juca,
1989; Lu, 1992; Shen and Yu, 1996; Xu, 1997).
Recently, soft computing techniques, such as Artificial
Neural Network (ANN), Fuzzy System, Genetic Expression Programming, and others, have been used frequently
to solve a wide variety of problems in geosciences and
geotechnical engineering. These include the estimation of
the probability of liquefaction (Youd and Gilstrap, 1999;
Juang et al., 2000; Goh et al., 1994; Hanna et al., 2007),
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Front. Earth Sci. 2014, 8(3): 439–456
strength parameter modeling of different soils (Agrawal et
al., 1994; Attoh-Okine, 2004; Goktepe et al., 2008; Kaya,
2009), hydraulic conductivity (Akbulut, 2005), predicting
the angle of internal friction in soils using a hybrid genetic
fuzzy system (Goktepe and Sezer, 2010), identification of
compaction characteristics (Najjar and Basheer, 1996;
Alavi et al., 2009), and the problem of slope stability
(Ferentinou and Sakellariou, 2007; Cho, 2009). Kayadelen
et al. (2009) studied the prediction of the angle of internal
friction of soils using soft computing techniques.
Among all of these soft computing techniques, ANN has
a wide range of applicability in cases of landslide
susceptibility zonation (LSZ), prediction of debris flow,
landslide movement monitoring, prediction of strength and
deformation properties of rock, etc. Lu and Rosenbaum
(2003) built models using artificial neural networks and
grey systems for the prediction of slope stability. Neaupane
and Achet (2004) used a back propagation neural network
for monitoring a landslide in the higher Himalaya. ANN
models (Arora et al., 2004; Gómez and Kavzoglu, 2005;
Yesilnacar and Topal, 2005; Kanungo et al., 2006;
Nefeslioglu et al., 2008) have been implemented for LSZ
studies. Sonmez et al. (2006) used ANN for the
determination of the deformation modulus of intact rock
specimens. Das and Basudhar (2008) used artificial neural
networks for the prediction of the residual friction angle of
clays. Tiryaki (2008) used multivariate statistics, artificial
neural networks, and regression trees to predict intact rock
strength and deformation properties for mechanical
excavations. Baykasoğlu et al. (2008) used genetic
programming for the prediction of compressive and tensile
strength of Gaziantep limestone. Maji and Sitharam (2008)
used the ANN model for the prediction of elastic modulus
of jointed rock mass. Çanakcı et al. (2009) predicted the
compressive and tensile strength of Gaziantep basalts
using neural networks and gene expression programming.
Rafiai and Jafari (2011) developed a new set of rock failure
criteria using the ANN approach.
In the present study, indirect estimation of shear strength
parameters, such as friction angle (f) and cohesion (c) of
soil, under unsaturated conditions was done using ANN
and Regression Tree (CART) approaches.
2
Data used for the study
Laboratory testing of soils from surface and subsurface
areas is a fundamental element of the geotechnical
investigation of a site before the design and practice of
any civil engineering construction. These laboratory tests
may vary from simple soil classification tests to complex
strength and deformation tests. In this research, soil
samples have been obtained from both surface and
subsurface soil resources from 6 different states of India,
namely, Himachal Pradesh, Uttar Pradesh, Bihar, Jharkhand, Orissa, and Andhra Pradesh. A series of laboratory
tests were conducted to determine the engineering properties of these soil samples. All tests were performed
according to Indian Standard 2720 (IS 2720: Parts IV, V
and XIII). The tests performed included the grain size
distribution, Atterberg limits, dry density, and direct shear
test. Soil parameters including gravel % (GP), sand % (SP),
silt % (STP), clay % (CP), dry density (DD), plasticity
index (PI), c, and f were measured on 115 samples. These
data were taken from the unpublished reports of the
Institute and utilized for this research work. The basic idea
in this research is the evaluation of the capability of ANN
and CART techniques to make indirect estimates of the
shear strength parameters of soils under unsaturated
conditions.
Statistical descriptions of the soil parameters of all 115
soil samples are given in Table 1. It can be seen from this
table that the median and average values of each parameter
are similar. This shows that the statistical distributions of
each parameter for all of the soil samples are nearly
normal. As shown in Table 1, the measured values of c
range from 0.0 to 0.7 kg/cm2, with an average value of
0.14 kg/cm2, and a median value of 0.08 kg/cm2. The f
values range from a lower value of 9 degrees to a higher
value of 40.5 degrees. The mean and median values of f
are 25.4 and 26 degrees respectively. Amongst all of the 6
soil parameters, DD has the least spatial variation; and SP
has the largest spatial variation.
A correlation matrix was produced by applying a
bivariate correlation technique to the original data set, in
order to analyze the strength of the linear relationships
Table 1 Basic descriptive statistics for different parameters of soil samples
Statistics
GP
SP
STP
CP
DD
/(gm$cm–3)
PI
c/(kg$cm–2)
f /degree
Minimum
0
5
0
0
1.24
0
0
9
Maximum
59
97
82
48
2.04
45.22
0.7
40.5
Average
4.456522
59.56957
25.84783
10.24783
1.838817
10.23383
0.139087
25.40261
Median
0
61
23
10
1.9
11
0.08
26
8.730257
23.65652
17.82655
10.6948
0.166569
9.609107
0.164424
7.200542
115
115
115
115
115
115
115
115
Standard
deviation
Number of
samples
D.P. KANUNGO et al. ANN & CART for prediction of soil shear parameters
between the variables considered in this case. The
correlation coefficients (R) values between c and f, the
dependent variables, and the other 6 soil properties (GP,
SP, STP, CP, DD, and PI), the independent variables, were
investigated. The correlation coefficient matrix for all soil
parameters, along with their significance levels, is given in
Table 2. The results in Table 2 show that SP, STP, and CP
have influences on c. STP and CP are significant factors
contributing to c, with positive R values of 0.589 and
0.512, respectively. SP also appears to influence c, but with
a negative R of –0.672. As per the significance levels of
different soil parameters (refer to Table 2), GP is the only
insignificant parameter with respect to c. Also SP, STP, CP
and PI have influences on f. SP is the most significant soil
parameter contributing to f of soil, with a positive R value
of 0.661. STP, CP, and PI are related to f with negative R
values of –0.651, –0.603 and –0.631, respectively. It is also
observed in Table 2 that DD is the only insignificant
parameter with respect to f.
3
Methodology
3.1
Artificial Neural Network (ANN)
ANN was originally devised as models of the human brain.
It was hoped that ANN could reveal useful information
about the structure of the brain and the processes that occur
within the brain. The use of ANN as a tool for exploring
brain function has become increasingly widespread within
441
cognitive psychology and neurophysiology in recent years.
However, this study is primarily interested in ANN as a
tool for solving mathematical problems. In particular,
ANN is used to identify unknown multivariate functions
from samples of data. Aspects concerning the biological
validity of ANN architecture, or of a training algorithm, are
only occasionally considered.
Neural networks are non-linear statistical data modeling
tools. They can be used to model complex relationships
between inputs and outputs, or to find patterns in data. A
set of connected units forms a neural network with the
capability of nonlinear input/output approximations. If the
units are grouped into layers, and all units of a layer are
connected with the units of the subsequent layer, a feed
forward network is developed (i.e., a static process).
Each layer in a network contains a sufficient number of
neurons depending on its specific application. The neurons
in a layer are connected to the neurons in the next
successive layer; and each connection carries a weight
(Atkinson and Tatnall, 1997). The input layer receives the
data from different sources. Hence, the number of neurons
in the input layer depends on the number of input data
sources. The hidden and output layers actively process the
data. The number of hidden layers and their neurons are
often determined by trial and error (Gong, 1996). The
number of neurons in output layers is fixed by the
application. Each hidden neuron responds to the weighted
inputs it receives from the connected neurons from the
preceding input layer (Lee et al., 2004). Once the
combined effect on each hidden neuron is determined,
Table 2 Correlation matrix and significance levels for the considered data set
Correlation matrix
Parameters
GP
GP
SP
STP
CP
DD
PI
c
f
1
–0.102
–0.159
–0.327
–0.252
–0.182
–0.012
0.274
1
–0.890
–0.648
0.223
–0.679
–0.672
0.661
1
0.433
–0.275
0.531
0.589
–0.651
1
0.183
0.779
0.512
–0.603
1
0.006
–0.298
0.019
1
0.470
–0.631
1
–0.678
SP
STP
CP
DD
PI
c
f
1
Significance Levels
(Correlation is significant at the 0.01 level)
GP
SP
STP
CP
DD
PI
c
f
.
0.277
0.090
0.000
0.007
0.052
0.900
0.003
.
0.000
0.000
0.016
0.000
0.000
0.000
.
0.000
0.003
0.000
0.000
0.000
.
0.050
0.000
0.000
0.000
.
0.948
0.001
0.838
.
0.000
0.000
.
0.000
.
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Front. Earth Sci. 2014, 8(3): 439–456
the activation at this neuron is determined via a transfer
function (Yesilnacar and Topal, 2005). Any differentiable
nonlinear function can be used as a transfer function; but a
sigmoid function is generally used, though there are many
other functions (Schalkoff, 1997). The sigmoid function
constrains the outputs of a network to a range between 0
and 1.
This type of feed forward network propagates the input
vector Xmn from the input-layer, through one or more
hidden layers, to the output-layer, only in one direction. A
dynamic process can be carried out by adding external
feedback of delayed outputs; and this is referred to as
external recurrent networks. The relationship between the
input vector (Xmn ) and output vector (Xjnþ1 ) of this element
can be described as follows:
!
X
n n
Wjm
Xm ,
Xjnþ1 ¼ F
(1)
i
1
or
1 þ e–x
other nonlinear transfer function, e.g., tan-sigmoid funcn
tion, and Xjnþ1 is output of unit j in the nth layer, and Wjm
is
th
th
a weight from unit m in n layer to unit j in (m + 1) layer,
as shown in the Fig. 1. Network training is a process by
which the connection weights and biases of the ANN are
adapted through a continuous process of simulation by the
environment in which the network is embedded. The
primary goal of training is to minimize an error function,
by searching for a set of connection strengths and biases
that cause the ANN to produce outputs that are equal or
close to targets. The network connection strengths are
adjusted in the training process, which can be executed
through a number of learning algorithms based on back
propagation learning (Ripley, 1996; Haykin, 1998; Zhou,
1999; Lee et al., 2004; Gómez and Kavzoglu, 2005;
Yesilnacar and Topal, 2005). The most widely used back
propagation algorithms are gradient descent and gradient
descent with momentum. These are often too slow for the
solution of practical problems. The faster algorithms use
standard numerical optimizers such as conjugate gradient,
quasi-Newton, and Levenberg-Marquardt approaches. In
where F(x) is the log sigmoid function FðxÞ ¼
this study, the Levenberg-Marquardt algorithm (implemented as TRAINLM in MATLAB software) was used for
training the neural network. The details of this algorithm
can be found in Hagan and Menhaj (1994) and Hagan et al.
(1996). In other words, training aims at estimating the
parameters’ weights and biases by minimizing an error
function of the output values. The total sum squared error
E is averaged over all patterns in the training set, in which
Mp is the target output (predicted) for the pth pattern, and
Op is the actual output (measured).
Xp
E¼
ðMp – Op Þ2 :
(2)
p¼1
The process of back propagating the error is repeated
iteratively until the error is minimized to an acceptable
value and the adjusted weights are obtained, which are then
used to determine the network outputs. The performance of
the network depends on the accuracy obtained over a set of
test data. If the network is trained to an acceptable level of
accuracy, then the adjusted weights are used to determine
the outputs of the test data set.
The R and Root Mean Square Error (RMSE) were
considered for evaluating the ability of trained and tested
networks in predicting shear parameters. The coefficient of
determination is a measure of the accuracy of prediction of
the trained network models. Higher R values indicate better
prediction. In addition, the mean relative percentage error
was also used to measure the accuracy of prediction. For
each network, the training and testing errors/accuracies for
the individual output shear strength parameters (i.e., either
c or f) were calculated using Eq. (3) and Eq. (5).
Accuracies for multiple output shear strength parameters
(i.e., both c and f) were calculated using Eq. (4) and
Eq. (6).
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 Xn
RMSE ¼
ðMj – Oj Þ2 ,
(3)
j¼1
n
RMSEcombined
!
rffiffiffiffiffiffi n
n
X
1 X
2
2
¼
ðM – Oc Þ þ
ðMf – Of Þ ,
2n c¼1 c
f¼1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
!
u
u
ðMj – Oj Þ2
j
X
R ¼ t1 –
,
2
ðO
Þ
j
j
(4)
(5)
XX
Mjj – M Ojj – O
j
j
ffi
, (6)
Rcombined ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
XX
2 X X 2
Mjj – M
Ojj – O
j
Fig. 1 A schematic description of the relationship between the
input and output vectors of one neuron.
j
j
j
where j = number of data patterns in the independent data
set; M = actual data set; O = predicted data set; M = mean
D.P. KANUNGO et al. ANN & CART for prediction of soil shear parameters
of actual data set; O = mean of predicted data set.
In this study, a feed forward back-propagation (FFBP)
multi-layer ANN, with one input layer, one hidden layer,
and one output layer was used to predict the shear strength
parameters of the soil. The input layer contains a maximum
of 6 neurons, each representing a parameter (GP, SP, STP,
CP, DD, and PI) that contributes to the shear strength of the
soil. The output layer contains two different neurons,
representing c, and f of soil under an unsaturated condition
for a given set of input values. It is important to ensure that
the training samples given to the network cover the range
of possibilities and are representative of the frequency of
occurrences of the variables. The Levenberg-Marquardt
algorithm (implemented as TRAINLM in MATLAB
software) was used for training the neural network; and
the log-sigmoid function was applied for neurons between
the input and hidden layers, and also for neurons between
the hidden and output layers. By varying the number of
neurons in the hidden layer, the neural networks are run
several times to identify the most appropriate neural
network architecture based on training and testing
accuracies. The neural network processing that was
implemented, using coding developed in MATLAB Software and the computation procedure of the ANN process,
is given in Fig. 2. In this study, 95 of the simulated samples
in a random order formed the training data; while the
remaining 20 simulated samples in a random order were
used for both validation and testing.
3.2
443
Regression tree
In recent times, there has been increasing interest in the use
of classification and CART analysis. CART analysis is a
tree-building technique, which is different from traditional
data analysis methods. Breiman et al. (1984) developed
CART, which is a sophisticated program for fitting trees to
data. Depending on the value of the predictor variables, a
decision tree partitions the data set into regions, so that the
response variable is roughly constant in that region. The
attractive feature of the CART methodology is a sequence
of hierarchical questions that the algorithm asks. This
method is relatively simple to understand and interpret.
The questions can be answered as ‘yes’ or ‘no’; and
depending on the answer, it either proceeds to another
question, or arrives at a fitted response value.
Details about the mathematical treatment of regression
tree analysis are discussed in Breiman et al. (1984). The
unique starting point of a classification tree is called a root
node and consists of the entire learning data set at the top of
the tree. A node is a subset of the set of variables, and it can
be a terminal or non-terminal node. A non-terminal node is
a node that splits into two daughter nodes (binary split).
Such a binary split is determined by a condition on a single
variable, where the condition is satisfied or not satisfied by
the observed value of that variable. All observations in the
whole dataset that have reached a particular node and
satisfy the condition for that variable drop down to one of
the daughter nodes. The remaining observations at that
node that do not satisfy the condition drop down to the
other daughter node. A node that does not split is called a
terminal node. In each terminal node, t, the fitted value, or
the predicted response value y(t) is constant. Suppose the
data points (x1,y1), (x2,y2),…,(xn,yn) are all of the samples
belonging to one terminal node, say node l. Then the
simple model for l is given in Eq. (7), the sample mean of
the dependent variable in that cell.
1 X
yðlÞ ¼
y:
(7)
N ðtÞ x 2 l n
n
Since the predictor, yðlÞ, is constant over each terminal
node, the regression tree can be thought of as a histogram
estimate
of the regression surface. For every node l,
X
ðyn – yðlÞÞ2 is the within node sum of squares, that is
xn 2l
Fig. 2 Neural network processing as implemented using coding
developed in MATLAB software.
the total squared deviations of yn in l from their average.
Summing over the entire terminal node (l2T) gives the
total within node sum of squares; and dividing by N gives
the average (Eq. 8).
1 XX
E¼
ðy – yðlÞÞ2 :
(8)
N l2T x 2l n
n
Therefore, a particular regression tree is formed by
iteratively splitting nodes to maximize the decrease in E
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Front. Earth Sci. 2014, 8(3): 439–456
(Breiman et al., 1984). There are two methods to grow a
decision tree: (a) stop recursive partitioning when the
largest decrease in E would be less than some user defined
threshold value say δ, and (b) each terminal node should
not contain more or equal to P sample data points, where P
is a user defined integer number (splitmin). In this study,
the second method is used to grow the decision trees.
4 Implementation of different techniques
for the prediction of shear strength
parameters of soil
The estimation of unsaturated shear strength parameters of
soils was performed using both ANN and CART
techniques. Both of the techniques used different combinations of GP, SP, STP, CP, DD, and PI as the input
parameters, and the unsaturated c, and f of soil as the
target outputs. Though GP is insignificant with respect to c,
and DD is insignificant with respect to f as per the
correlation matrix of the original data set (refer to Table 2),
neither of these two parameters (GP and DD) could be
excluded from the ANN models. This is because of the fact
that this study attempts to develop a single ANN model for
estimating both c and f, considering 6 input parameters
(GP, SP, STP, CP, DD, and PI) in combination. Secondly, in
order to cross verify the inference of the correlation matrix
with the capability of the ANN models through the
connection weight and bias analysis for input parameter
importance for estimating c and f, all 6 parameters in
combination are considered for the ANN models.
With different combinations out of these 6 input
parameters, four different models were tried to establish
the efficacy of different input combinations for predicting
the shear strength parameters of soil. These models are as
follows: (a) Model I: GP, SP, STP, and CP; (b) Model II:
GP, SP, STP, CP, and DD; (c) Model III: GP, SP, STP, CP,
and PI; and (d) Model IV: GP, SP, STP, CP, DD, and PI.
These techniques were implemented with MATLAB
software to predict the unsaturated shear strength parameters of the soil samples.
4.1
Prediction using ANN technique
A feed forward back-propagation multi-layer ANN with
one input layer, one hidden layer, and one output layer was
considered in the present study. The number of neurons in
the input layer equals the number of input parameters
considered in case of each model (i.e., 4 inputs in the 1st
case, 5 inputs in each of the 2nd and 3rd cases, and all 6
inputs in the 4th case). The data at each neuron of the input
layer correspond to the normalized value of each input.
The input parameters GP, SP, STP, and CP are normalized
with respect to 100 as these represent the percent value.
The input parameters DD and PI are normalized with
respect to their maximum value of occurrence (i.e., 2.04 in
the case of DD and 45.22 in the case of PI). The output
layer consists of two different neurons, representing the
experimentally determined value of unsaturated c, and f of
soil as the target outputs. The values of c and f are also
normalized with respect to their maximum values of
occurrence (i.e., 0.7 kg/cm2 and 40.5 degrees respectively).
The number of neurons in the hidden layers is varied by
running the networks several times to achieve the desired
training and testing data accuracies.
One set each of training and testing data were randomly
generated from the available data set using the stratified
random sampling technique. The training dataset consists
of data pertaining to 95 soil samples and the testing dataset
consists of data pertaining to 20 soil samples out of a total
of 115 samples. All of the data points in the datasets were
mutually exclusive (Foody and Arora, 1997). The training
dataset was used to train different network architectures,
while the testing dataset was used simultaneously with the
training dataset to control the overtraining of the network.
The testing dataset was also used to evaluate the accuracy
of the networks. The well-known back propagation
Levenberg-Marquardt algorithm was used to train the
neural networks. 40 neural network architectures were
created by varying the number of neurons in the hidden
layer, and then trained and tested. The training process was
initiated with arbitrary initial connection weights, which
were constantly updated until an acceptable level of
accuracy was reached. The final adjusted weights of the
trained network were used to derive outputs of the testing
data to evaluate the performance of the network.
4.2
Prediction using the regression tree technique
A number of regression trees were calculated, respectively,
for c and f, using MATLAB and employing different
combinations of input parameters (GP, SP, STP, CP, DD,
and PI) as predictors. The original experimentally
determined values of all of the parameters were used for
the analysis. Determining the right size tree is not that
straight forward. 50 different trees with varying splitmins
were built for each individual combination of predictors.
The performances of the trees developed in this study were
assessed using two different statistical performance
evaluation criteria. The statistical measures used were the
R and RMSE, the same as those adopted in the ANN
technique.
5
Results and discussion
5.1
Results of ANN technique
5.1.1
Performance evaluation of neural networks
The performance of the networks was evaluated by
determining both the training and testing data accuracies
D.P. KANUNGO et al. ANN & CART for prediction of soil shear parameters
in terms of R and RMSE values (Freund, 1992). The
combined accuracies in terms of R and RMSE values for
both of the shear parameters (c and f) were calculated with
MATLAB software for each of the four models, using
different combinations of input parameters (as mentioned
in Section 4). The training and testing accuracies for all of
the 4 models for some selected networks out of a total 40
neural networks with neurons in the hidden layer varying
from 1 to 40 are given in Table 3.
A variation in both training and testing data accuracies
across different models can be seen as the neural network
architectures change. The training and testing accuracies in
445
terms of R values observed across different neural
networks for different models were as follows: (a) Model
I: –0.05 to 0.99 for training and 0.12 to 0.82 for testing, (b)
Model II: 0.13 to 0.99 for training and –0.09 to 0.91 for
testing, (c) Model III: 0.76 to 0.99 for training, and 0.48 to
0.81 for testing and (d) Model IV: 0.13 to 0.99 for training
and 0.27 to 0.88 for testing. This implies that there an
optimal network architecture for a given dataset exists.
In the present case, the network architectures 4/6/2, 5/
16/2, 5/18/2, and 6/2/2 were observed to be the most
appropriate ones for Models I, II, III, and IV, respectively,
with the training and testing data accuracies, as highlighted
Table 3 Combined accuracies in terms of R and RMSE’s for both the shear parameters (c and f ) for all 4 models for some selected neural networks
R
ANN architecture
RMSE
Training
Testing
Training
Testing
0.904788
0.709213
0.128307
0.202459
Model I (GP, SP, STP and CP as inputs)
4/1/2
4/6/2
0.950486
0.820566
0.094369
0.094369
4/7/2
0.938912
0.73586
0.103917
0.203841
4/20/2
0.983056
0.723992
0.056938
0.229616
4/24/2
0.98938
0.709713
0.045527
0.226808
4/32/2
0.984646
0.724235
0.052902
0.215475
4/40/2
0.991034
0.58296
0.040481
0.27849
5/1/2
0.894149
0.779981
0.134873
0.177807
5/3/2
0.894153
0.894153
0.134871
0.177657
5/6/2
0.933653
0.85501
0.108095
0.15316
Model II (GP, SP, STP, CP and DD as inputs)
5/16/2
0.978625
0.915253
0.062174
0.121163
5/19/2
0.993896
0.861629
0.033275
0.16182
5/32/2
0.995472
0.886153
0.02864
0.135251
5/40/2
0.890366
0.671108
0.282153
0.37535
Model III (GP, SP, STP, CP and PI as inputs)
5/1/2
0.887056
0.735596
0.139076
0.192906
5/7/2
0.967071
0.773272
0.076868
0.201282
5/13/2
0.98884
0.800253
0.047095
0.19212
5/18/2
0.988724
0.810958
0.045282
0.180246
5/21/2
0.986577
0.804543
0.04952
0.19896
5/29/2
0.993524
0.779988
0.034499
0.20616
5/40/2
0.996953
0.639055
0.023531
0.2481
6/1/2
0.900495
0.801914
0.130986
0.170389
6/2/2
0.939658
0.879199
0.103081
0.136468
6/5/2
0.974072
0.87192
0.069304
0.147565
Model IV (GP, SP, STP, CP, DD and PI as inputs)
6/7/2
0.98054
0.877744
0.059174
0.156894
6/15/2
0.975545
0.861556
0.066319
0.16628
6/22/2
0.996851
0.848852
0.023911
0.169109
6/40/2
0.999205
0.761987
0.012016
0.216977
446
Front. Earth Sci. 2014, 8(3): 439–456
in Table 3 for the prediction of shear parameters of soil
under unsaturated conditions. For the above most appropriate network architectures, the training R values across
all of the models varied from 0.94 to 0.99, whereas the
testing R values across these four models varied from 0.81
to 0.91. It can be inferred from these results that there
exists a good correlation between training and testing
results along with the good prediction capability achieved
by the neural networks with the present data set.
If the results of all of the four models are compared for
the prediction of c and f, it can be observed from Table 3
that Model II, considering 5 input parameters such as GP,
SP, STP, CP, and DD, has given the highest accuracies (5/
16/2 neural network). The training and testing R values as
obtained for Model II for the 5/16/2 neural network are
illustrated in Fig. 3. However, in the case of Model IV,
considering all of the 6 input parameters, the difference
between training and testing R values (6/2/2 neural
network) is the least when compared to the other three
models (Kanungo et al., 2006). The training and testing R
values obtained for Model IV for the 6/2/2 neural network
are illustrated in Fig. 4. If several networks fit the training
set equally well, then the simplest network (i.e., the
network that has the smallest number of weights and
biases) will, on average, give the best generalization
performance (Sietsma and Dow, 1991). Hence, it can be
stated from these observations that Model IV is the most
appropriate model for the prediction of both the shear
parameters of soil under unsaturated conditions.
In order to assess the individual accuracy of each of the
shear parameters (c and f) separately instead of combined
(as discussed above) for all of the four models, the
predicted and the target output values of both c and f were
analyzed in each case for all the neural networks. The
training and testing accuracies in terms of R and RMSE
values for all of the 4 models for some selected networks
are given in Table 4.
A variation in both training and testing data accuracies,
for both c and f across different models, can be seen as the
neural network architectures change. The training and
testing accuracies in terms of R values observed across
different neural networks for different models in predicting
c were as follows: (a) Model I: –0.60 to 0.98 for training
and –0.60 to 0.75 for testing, (b) Model II: –0.31 to 0.99
for training and –0.34 to 0.92 for testing, (c) Model III:
–0.09 to 0.99 for training and –0.32 to 0.82 for testing and
(d) Model IV: –0.37 to 0.99 for training and –0.48 to 0.91
for testing (Fig. 5). Similarly, for f, the training and testing
accuracies in terms of R values observed across different
neural networks for different models were as follows: (a)
Model I: 0.0 to 0.97 for training and –0.41 to 0.80 for
testing, (b) Model II: 0.08 to 0.99 for training and –0.42 to
0.87 for testing, (c) Model III: –0.32 to 0.99 for training
and –0.32 to 0.89 for testing and (d) Model IV: –0.31 to
0.99 for training and –0.60 to 0.94 for testing (Fig. 6).
For the prediction of c, the network architectures 4/6/2,
5/16/2, 5/21/2, and 6/5/2 were observed to be the most
appropriate ones for Models I, II, III, and IV, respectively,
with the training and testing data accuracies as highlighted
in Table 4. For these most appropriate network architectures, the training R values across all of the models varied
from 0.90 to 0.97; whereas, the testing R values across
these four models varied from 0.75 to 0.91. It can also be
observed, from the results given in Table 4, that for the
Fig. 3 Correlation coefficients as obtained for Model II for the 5/16/2 neural network: (a) training and (b) testing.
D.P. KANUNGO et al. ANN & CART for prediction of soil shear parameters
447
Fig. 4 Correlation coefficients as obtained for Model IV for the 6/2/2 neural network: (a) training and (b) testing.
prediction of f of soil the network architectures 4/6/2, 5/
16/2, 5/27/2, and 6/8/2 were the most appropriate ones for
Models I, II, III, and IV, respectively. For these most
appropriate network architectures, the training R values
across all four models varied from 0.84 to 0.98; whereas,
the testing R values across these four models varied from
0.80 to 0.92. It can be stated from these results that there
exists a good correlation between training and testing
results even though the individual prediction of both of the
shear parameters of the soil is considered.
If the results of all of the four models are compared for
the prediction of c and f individually, Table 4 shows that
Model IV, which considers all 6 input parameters, is the
most appropriate model with the highest accuracies for the
prediction of both of the shear parameters of soil under
unsaturated conditions.
In summary, the overall results indicate that, whether the
training and testing data accuracies for the prediction of
both shear parameters for soil (c and f) are analyzed
combined or individually, it hardly affects the final result in
terms of the most appropriate model for prediction, Model
IV, which considers all 6 input parameters. But it is always
easy and convenient to adopt the combined way of
accuracy analysis for prediction of both c and f with
multiple neurons in the output layer (in this case 2 neurons
for c and f) to select the best single neural network
architecture.
5.1.2
Connection weight analysis
The ANN connection weights (Table 5) obtained for the
most appropriate neural network for the best model (6/2/2
neural network for Model IV using all 6 input parameters
in the case of combined analysis) to predict both c and f
have been used to characterize the input data sources (GP,
SP, STP, CP, DD, and PI) in terms of their influence on the
shear strength parameters of soil (c and f). In this process,
the connection weight approach (Olden and Jackson,
2002) and Garson’s Algorithm (Garson, 1991; Gevrey et
al., 2003) have been used to assess the influence of each
variable, in the most appropriate neural network (i.e., 6/2/
2), on the prediction of c and f. Both of these approaches
use connection weights between input-hidden and hiddenoutput neurons only and not the biases. The connection
weight approach calculates the product of the raw inputhidden and hidden-output connection weights between
each input neuron and output neuron, and sums the
products across all hidden neurons (Olden and Jackson,
2002; Olden et al., 2004). Garson’s algorithm partitions
hidden-output connection weights into components associated with each input neuron using the absolute values of
connection weights (Garson, 1991; Gevrey et al., 2003;
Olden et al., 2004). As in this case, the neural network with
6/2/2 architecture (representing 6 neurons in the input
layer, 2 neurons in the hidden layer and 2 neurons in the
output layer) has the connection weight matrices of 62,
and 22 size. The product of 62 and 22 matrices, using
the connection weight approach and Garson’s approach,
gives a resultant matrix of 62, which corresponds to the
weights of the 6 input variables. The values of these
weights are used to rank the input variables, meaning that
the variable with maximum weight is assigned as rank 1
and the variable with the minimum weight as rank 6. The
weights and ranks corresponding to all the 6 input
448
Front. Earth Sci. 2014, 8(3): 439–456
Table 4 Individual accuracies in terms of R and RMSE values for both of the shear parameters (c and f) for all the 4 models for some selected neural
networks
f
c
ANN architecture
R
Training
RMSE
Testing
R
RMSE
Training
Testing
Training
Testing
Training
Testing
0.336
1.181
0.808
0.693
0.079
0.516
Model I (GP, SP, STP and CP as inputs)
4/1/2
0.710
0.564
4/6/2
0.896
0.752
1.730
5.821
0.844
0.798
0.038
0.923
4/7/2
0.833
0.639
1.148
2.083
0.864
0.716
0.320
1.102
4/24/2
0.985
0.670
1.551
0.044
0.958
0.494
0.000
0.414
4/28/2
0.958
0.742
2.006
2.188
0.906
0.377
0.036
0.558
4/32/2
0.968
0.694
0.784
1.208
0.954
0.693
0.007
0.702
4/40/2
0.981
0.555
0.645
0.029
0.975
0.236
0.040
1.592
Model II (GP, SP, STP, CP and DD as inputs)
5/1/2
0.712
0.714
0.036
0.870
0.727
0.719
0.020
0.435
5/6/2
0.845
0.850
0.820
0.589
0.808
0.791
0.355
0.517
5/16/2
0.950
0.897
0.740
0.624
0.943
0.871
0.118
0.717
5/22/2
0.989
0.896
0.041
1.832
0.987
0.582
0.091
0.503
5/32/2
0.988
0.924
0.090
0.046
0.991
0.660
0.049
0.052
5/33/2
0.988
0.880
0.119
0.755
0.989
0.636
0.073
0.461
5/40/2
0.993
0.681
0.179
0.038
0.079
0.411
34.165
8.704
Model III (GP, SP, STP, CP and PI as inputs)
5/1/2
0.657
0.626
0.305
0.888
0.755
0.691
0.002
0.380
5/9/2
0.851
0.510
0.304
0.971
0.888
0.792
0.231
1.025
5/13/2
0.977
0.809
1.783
0.989
0.967
0.684
0.120
0.240
5/18/2
0.975
0.808
0.587
0.180
0.968
0.603
0.076
0.698
5/21/2
0.970
0.820
0.735
1.261
0.963
0.731
0.355
0.380
5/27/2
0.989
0.680
0.422
1.735
0.985
0.886
0.017
0.469
5/40/2
0.994
0.638
0.168
0.199
0.990
0.354
0.069
0.623
Model IV (GP, SP, STP, CP, DD and PI as inputs)
6/1/2
0.719
0.751
0.008
0.969
0.765
0.744
0.011
0.536
6/2/2
0.861
0.853
0.311
0.857
0.824
0.854
0.178
0.803
6/5/2
0.959
0.908
1.582
0.658
0.899
0.679
0.214
0.082
6/8/2
0.979
0.799
0.222
1.265
0.916
0.922
0.045
0.190
6/15/2
0.963
0.839
0.410
0.872
0.902
0.812
0.046
0.700
6/22/2
0.995
0.828
0.163
0.466
0.988
0.815
0.048
0.249
6/40/2
0.999
0.728
17.029
1.422
0.997
0.791
34.165
0.358
variables for the prediction of both c and f are given in
Table 6.
Ideally, the finer fractions (CP and STP), along with PI
of the soil material, contribute more towards its c than
coarser fractions (GP and SP) along with DD. Conversely,
the coarser fractions (GP and SP), along with DD of the
soil material, contribute more towards its f than the finer
fractions (CP and STP) along with PI. This is depicted in
the correlation matrix of the original data set (refer to Table
2) used for this study.
It is observed from Table 6 that, for the prediction of c,
the descending order of influence of the 6 different input
variables in the case of the connection weight approach is
PI, STP, SP, CP, DD, and GP. That in case of Garson’s
approach is CP, GP, PI, DD, SP, and STP. Similarly, from
Table 6, it is also observed that, for the prediction of f, the
descending order of influence of the 6 input variables in the
case of the connection weight approach is GP, DD, CP, SP,
D.P. KANUNGO et al. ANN & CART for prediction of soil shear parameters
449
Fig. 5 ANN training and testing accuracies in terms of correlation coefficients observed across different neural networks for the case of c
for Model IV.
Fig. 6 ANN training and testing accuracies in terms of correlation coefficients observed across different neural networks for the case of
f for Model IV.
Table 5 Connection weights and biases for cohesion (c) and friction angle (f) in case of a 6/2/2 neural network
Weights (wik)
Weights ( vko)
Biases
Neurons
Input 1
(GP)
Input 2
(SP)
Input 3
(STP)
Input 4
(CP)
Input 5
(DD)
Input 6
(PI)
Output 1 (c)
Output 2 (f)
bk
Hidden Neuron
1 (k = 1)
–124.98
–5.75
–0.98
20.12
–101.27
14.18
2.34
–1.02
87.64 –4.80
Hidden Neuron
2 (k = 2)
1.85
–4.18
2.66
–67.58
6.05
56.99
2.94
–1.59
–2.69
bo
(c)
—
(f)
2.10
—
450
Front. Earth Sci. 2014, 8(3): 439–456
Table 6 Weights and ranks (in brackets) corresponding to all of the 6 input variables for the prediction of both c and f as obtained using Connection
weight approach, Garson’s approach and Weight-bias approach
Weights and ranks
Input parameters
Connection weight approach
Garson’s apporach
Weight-bias approach
c
f
c
f
c
f
GP
–287.569167
(6)
125.21965
(1)
0.480890716
(2)
0.480890716
(2)
–41.045
(5)
12.532
(3)
SP
–25.7740852
(3)
12.54380092
(4)
0.051503799
(5)
0.051503799
(5)
–1.7203
(4)
22.123
(2)
STP
5.51500891
(2)
–3.22028038
(5)
0.02275593
(6)
0.02275593
(6)
–118.6067
(6)
83.4911
(1)
CP
–151.67199
(4)
86.88804508
(3)
0.560407326
(1)
0.560407326
(1)
200.5407
(3)
–88.731
(4)
DD
–219.64503
(5)
94.23460104
(2)
0.422294122
(4)
0.422294122
(4)
232.2052
(2)
–105.052
(5)
PI
200.9270384
(1)
–105.211826
(6)
0.462148106
(3)
0.462148106
(3)
366.723
(1)
–171.444
(6)
STP, and PI; and that in the case of Garson’s approach is
CP, GP, PI, DD, SP, and STP (exactly the same as in the
case of c). This result illustrates, to some extent, the critical
difference between the connection weight approach and
Garson’s approach that results in their differential ability to
correctly identify variable importance in the neural
network.
The connection weight product matrix (Table 6) shows
that the finer fractions (CP and STP), along with PI, have
comparatively more influence than a coarser fraction (GP),
along with DD, on the prediction of c, as ideally the case
should be. However, there exists an ambiguity in this case.
SP, being a coarser fraction of soil, exhibits more influence
on the prediction of its c than the finer fraction CP.
Similarly, it is also demonstrated in the connection weight
product matrix (Table 6) that the coarser fractions (GP),
along with DD, have comparatively more influence than
the finer fractions (CP and STP), along with PI, on the
prediction of f of the soil, as ideally the case should be.
However, there exists a similar ambiguity as discussed
above. On the other hand, the inability of Garson’s
algorithm to correctly depict the true order of variable
importance can be simply because it uses absolute
connection weights in its calculations, and fails to account
for the contrasting influences of input neurons through
different hidden neurons to output neurons. In this study,
the Garson’s weight product matrices for both c and f are
exactly same.
It is also an interesting finding that, in the case of
Garson’s algorithm for weight analysis, no contrast in the
order of influence of input variables is depicted in the case
of neural network architectures with multiple output
neurons (as in this case two output neurons c and f). In
contrast, the connection weight approach uses raw
connection weights, which accounts for the direction of
the input–hidden–output relationship and results in the
correct identification of variable contributions.
5.1.3
Connection weight-bias analysis
In the present study, an attempt was made to develop an
approach that takes into account both the connection
weights between input – hidden and hidden – output neurons, and biases at hidden and output neurons (Table 5), in
order to assess the influence of input variables in the most
appropriate neural network (i.e., 6/2/2) on the prediction of
c and f. According to Goh et al. (2005), a model equation
can be established with these weights and biases as the
model parameters to obtain the network output. The
mathematical equation relating the input variables and the
output can be written as:
(
"
h
X
cn or fn ¼ flog-sig b0 þ
vko flog-sig
k¼1
bk þ
m
X
!#)
wik Xi
,
(9)
i¼1
where cn or fn is the normalized (in the range of 0 to 1 in
this case) value of c or f depending on the weights and
biases; b0= bias at the output layer; vko connection weight
between kth neuron of hidden layer and the output layer (o);
bk= bias at the kth neuron of hidden layer; h = number of
neurons in the hidden layer; wik= connection weight
between ith input neuron and kth neuron of hidden layer;
Xi= normalized input variable i in the range [0,1]; m = total
No. of input
variables and
flog-sig= log-sigmoid transfer
1
function f ðxÞ ¼
.
1 þ e–x
Using the values of the weights and biases presented in
D.P. KANUNGO et al. ANN & CART for prediction of soil shear parameters
Table 5, one can easily calculate the normalized values of
unsaturated c and f respectively, which need to be
denormalized with respect to their maximum values in
the original data set.
Taking a lead from the above Eq. (9), the following
equations (Eqs. (10) and (11)) are designed to calculate the
resultant matrix of 62, which corresponds to the weights
of 6 input variables for the prediction of both c and f, using
connection weights as well as biases (named the Weightbias Approach).
8
9 8
9
b0 ðcÞ þ v11 ðb1 þ w11 Þ þ v21 ðb2 þ w21 Þ >
GP >
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
b
ðcÞ
þ
v
ðb
þ
w
Þ
þ
v
ðb
þ
w
Þ
>
>
>
SP
0
11 1
12
21 2
22 >
>
>
>
>
>
>
>
>
>
>
> >
>
>
>
< STP = < b0 ðcÞ þ v11 ðb1 þ w13 Þ þ v21 ðb2 þ w23 Þ >
=
¼
,
>
>
CP >
b0 ðcÞ þ v11 ðb1 þ w14 Þ þ v21 ðb2 þ w24 Þ >
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
DD
b
ðcÞ
þ
v
ðb
þ
w
Þ
þ
v
ðb
þ
w
Þ
>
>
>
>
0
11
1
15
21
2
25
>
>
>
>
>
>
>
:
:
; >
;
PI
b0 ðcÞ þ v11 ðb1 þ w16 Þ þ v21 ðb2 þ w26 Þ
(10)
9
8
9 8
b0 ðfÞ þ v12 ðb1 þ w11 Þ þ v22 ðb2 þ w21 Þ >
GP >
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
b
ðfÞ
þ
v
ðb
þ
w
Þ
þ
v
ðb
þ
w
Þ
>
>
>
SP
0
12 1
12
22 2
22 >
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
< STP = < b0 ðfÞ þ v12 ðb1 þ w13 Þ þ v22 ðb2 þ w23 Þ >
:
¼
>
> b0 ðfÞ þ v12 ðb1 þ w14 Þ þ v22 ðb2 þ w24 Þ >
>
> CP >
> >
>
>
>
>
>
>
> >
>
>
>
>
> DD >
> >
>
>
>
>
b
ðfÞ
þ
v
ðb
þ
w
Þ
þ
v
ðb
þ
w
Þ
>
>
>
0
12 1
15
22 2
25 >
>
>
>
>
>
>
>
:
;
:
; >
PI
b0 ðfÞ þ v12 ðb1 þ w16 Þ þ v22 ðb2 þ w26 Þ
(11)
The weights and ranks corresponding to all of the 6 input
variables for the Prediction of both c and f, as obtained
using the Weight-bias Approach, are also given in Table 6.
Here, it is observed that for the prediction of c, the
descending order of influence of 6 input variables in this
case is PI, DD, CP, SP, GP, and STP. For the prediction of
f, the descending order of influence of these variables is
STP, SP, GP, CP, DD, and PI. The resultant matrix (Table 6)
indicates that the finer fractions (CP) along with PI have
comparatively more influence than the coarser fractions
(SP and GP) on the prediction of c, as ideally the case
should be. However, there exist ambiguities in that STP,
being a finer fraction of soil, exhibits less influence on the
prediction of its c than the coarser fractions GP and SP.
Also, DD has more influence in predicting c, which should
not be the case.
It can be observed that the coarser fractions (SP and GP)
have comparatively more influence than the finer fractions
(CP) along with DD of the soil on the prediction of its f, as
the case should be. However, there also exist ambiguities
in this case; in that, STP being a finer fraction of soil
exhibits more influence on the prediction of its f than the
coarser fractions GP and SP. Also, PI has more influence in
451
predicting f, which again should not be the case. Hence, in
the case of the Weight-bias Approach, it can be stated that
the results to a great extent match the correlation matrix of
the original data set (refer to Table 2). This clearly reflects
the use of raw connection weights and biases for
generating the resultant matrix, as in the case of the
Connection Weight Approach.
5.1.4 Comparison of Connection Weight and Weight-bias
approaches
As the results for the Connection Weight Approach,
Garson’s Approach, and the proposed Weight-bias
Approach are compared, it can be summarized that, for
the assessment of degree of influence of input variables on
the prediction of shear strength parameters of soils under
unsaturated conditions, the most appropriate weight
analysis approach is the Connection Weight Approach. It
was found that the Connection Weight Approach provides
the best overall methodology for accurately quantifying
variable importance, and should be favored over the other
approaches examined in this study. The proposed Weightbias Approach performed relatively well. Garson’s
Approach showed both poor accuracy and precision, and
showed no contrast in weights across multiple outputs. The
most notable result of this study is that Garson’s Algorithm
is the poorest performing approach.
5.2
Results of the Regression Tree technique
As in the ANN technique, experimentally determined c
and f values of soils under unsaturated conditions are also
used as the target outputs in the regression tree technique.
Both the experimentally determined and observed values
of c and f are compared using R and RMSE measures in
the training and testing phase to assess the prediction
capability of this technique. The variation in RMSE and R
values across different trees with varying splitmins are
given in Fig. 7 for Model IV, with all 6 input parameters, to
assess their prediction capability for both c and f of soils.
The best prediction tree in each combination case is chosen
on the basis of these performance measures. Tables 7 and 8
contain the statistical performance details (RMSE and R) of
the best prediction trees for c and f, respectively, for four
different models. It can be inferred from the results given in
both tables that for different combinations of input
parameters the splitmin value varies for the best regression
tree for prediction of both c and f.
The results shown in Table 7 indicate that the best result
for the prediction of c has been obtained using Model IV,
which considers all 6 input parameters (GP, SP, STP, CP,
DD, and PI). The training and testing R values are 0.88 and
0.73 respectively. Also, the RMSE values for the training
and testing datasets are quite reasonable; and the difference
is less in this case than it is using the other 3 combinations
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Front. Earth Sci. 2014, 8(3): 439–456
Fig. 7 Variation in RMSE and R values across different regression trees with varying splitmins for Model IV for: (a) cohesion and (b)
friction angle.
Table 7 Training and testing accuracies for prediction of c using Regression Tree technique
RMSE/(kg$cm–2)
Predictors
Splitmin
R
Training
Testing
Training
Testing
GP,SP,STP, and CP
(Model I)
19
0.080
0.192
0.84
0.45
GP,SP,STP,CP, and DD
(Model II)
12
0.056
0.165
0.93
0.67
GP,SP,STP,CP, and PI
(Model III)
10
0.075
0.189
0.87
0.48
GP,SP,STP,CP,DD, and PI
(Model IV)
23
0.069
0.162
0.88
0.73
Table 8 Training and testing accuracies for prediction of f using Regression Tree technique
RMSE/degree
Predictors
Splitmin
R
Training
Testing
Training
Testing
GP,SP,STP, and CP
(Model I)
11
2.709
6.078
0.92
0.64
GP,SP,STP,CP, and DD
(Model II)
9
2.361
3.749
0.94
0.89
GP,SP,STP,CP, and PI
(Model III)
15
2.804
5.645
0.91
0.70
GP,SP,STP,CP,DD, and PI
(Model IV)
9
2.349
3.833
0.94
0.90
of input parameters. In cases using the other 3 combinations of input parameters, it can be inferred from the results
that the prediction trees are over-trained, and could not
attain generalization capability because the difference
between training and testing R is quite large.
Results in Table 8 show that, for the prediction of f,
again, the best result was obtained with Model IV,
considering all 6 input parameters, with training and
testing R values of 0.94 and 0.90, respectively. These
values are comparatively higher than those in the case of
prediction of c (refer to Table 7). In the case of Model II,
with input parameters GP, SP, STP, CP, and DD, the
prediction accuracies in terms of R values for training and
testing are 0.94 and 0.89, respectively, and are of the same
order as those for Model IV. Hence, it can be inferred from
these results that the input parameter PI has little influence
on the prediction or estimation of f. In the cases of the
other two combinations of input parameters, it can be
stated that the regression trees were over-trained, and did
not attain generalization capability because the difference
between training and testing R values is quite large.
Hence, it can be summarized from the above results that
D.P. KANUNGO et al. ANN & CART for prediction of soil shear parameters
the regression tree analysis in the case of Model IV,
considering all 6 predictors (GP, SP, STP, CP, DD, and PI),
gives the most appropriate result for the prediction of both
c and f. The most appropriate regression trees obtained for
c and f for Model IV, using all 6 input parameters as
predictors, are given in Fig. 8. The experimentally
determined and predicted c and f values for the testing
dataset of 20 soil samples in the case of Model IV are
represented in Fig. 9. This figure clearly shows a good
correlation between the experimentally determined and
predicted values of both shear parameters of soil.
6
453
Conclusions
In this research, ANN and CART techniques for the
prediction of shear strength parameters (c and f) of soils
under unsaturated conditions was investigated. The
performance evaluation of these techniques was carried
out using R and RMSE measures. In this work, four
different models were adopted, considering different
combinations of 6 input parameters of soil such as GP,
SP, STP, CP, DD, and PI. For all four models, with varied
combinations of input parameters, both ANN and CART
Fig. 8 Most appropriate regression tree in the case of Model IV for predicting (a) cohesion and (b) angle of internal friction.
454
Front. Earth Sci. 2014, 8(3): 439–456
techniques were attempted and evaluated. The results
indicate that amongst these models, Model IV, considering
all 6 input parameters, is the most appropriate model for
the prediction of shear parameters with both ANN and
CART techniques.
Fig. 9 Experimentally determined and predicted values for the
testing dataset in case of Model IV using regression tree technique:
(a) cohesion and (b) angle of friction.
In the case of the ANN technique, the 6/2/2 neural
network architecture gave the best training and testing
accuracy, with combined R values of 0.939 and 0.879,
respectively; and with combined RMSE values 0.103 and
0.136, respectively, for prediction of both c and f.
Similarly, in cases using regression tree analysis, it was
found that, for Model IV, the regression tree with a splitmin
of 23 obtained the best training and testing accuracy, with
R values 0.88 and 0.73, respectively; and with RMSE
values 0.069 and 0.162, respectively, for the prediction of
soil c. For the prediction of f, the regression tree with a
splitmin of 9 gave the best training and testing accuracy,
with R values 0.94 and 0.90, respectively; and with RMSE
values 2.349 and 3.833, respectively. With regression trees
like those generated in this study, having many branches,
there is the possibility of over fitting the tree towards the
training data set, hence it loses its generalization capability.
Keeping this in view, simpler regression trees in this case
have been tried; and results with considerably good
accuracies have been obtained.
From the results obtained using ANN and CART
techniques, it can be observed that for the prediction of
f, the performances of both the techniques are of the same
order, which is indicated by the similar R values for both
training and testing samples. However, for the prediction
of c, the ANN technique performs better than the
regression tree technique. Hence, it can be concluded
from these results that the ANN technique, with the
Levenberg-Marquardt learning rule, is a comparatively
better approach than the CART technique for the
prediction of both of the shear parameters of the soils.
Further, the performance evaluation of the ANN
technique, for the indirect estimation of both of the shear
parameters (c and f) of soils, has also been attempted on an
individual shear parameter basis, to assess and compare the
combined prediction capability vis-à-vis individual prediction capability of the ANN technique. It was observed
from the results that the individual or combined performance evaluation of both training and testing datasets did
not affect the overall result in selecting the most
appropriate model (Model IV considering all 6 input
parameters in this case) for the prediction of both of the
shear parameters of soil (c and f). However, it does have
effects in terms of the R values, and also in terms of the
most appropriate neural network architecture. So, it can
finally be stated that a combined performance evaluation is
always convenient to use in order to select the best single
neural network architecture for the prediction of both c
and f.
Connection weight and bias analyses of the best neural
network (i.e., 6/2/2) were attempted using the Connection
Weight Approach, Garson’s Approach, and a proposed
Weight-bias Approach to characterize the input variables in
terms of their influence on the shear strength parameters of
soil (c and f). It can be summarized that the Connection
Weight Approach appeared to be the most appropriate and
accurate weight analysis approach for quantifying variables importance. This method should be favored over the
other approaches examined in this study. The Connection
Weight Approach uses raw connection weights, which
accounts for the direction of the input-hidden-output
relationship, and results in the correct identification of
variable contribution. The proposed Weight-bias Approach
performed relatively well; and the results were comparable
with the Connection Weight Approach results. The most
notable result of this analysis was that Garson’s Algorithm
was the poorest performing approach; and no contrast in
the order of influence of the input variables was depicted
in cases of neural network architecture with multiple
output neurons (as in those cases with two output neurons c
and f).
Acknowledgements The authors are grateful to the Director, CSIR-CBRI
D.P. KANUNGO et al. ANN & CART for prediction of soil shear parameters
for his kind permission to publish this work. We also thank the editors and
two anonymous reviewers for their valuable comments, which helped to
improve the quality of the paper.
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